Regardless of warm ups and all that, your hands generally shouldn't be falling asleep after only 15 minutes of playing. Make sure your posture and hand position and correct. You CAN injure yourself playing guitar the wrong way (permanently if it's bad enough). An incorrectly positioned hand can lead to tendinitis or other hand/wrist problems (even back problems if your posture is incorrect).
Just make sure you are playing correctly to avoid injury. For me personally, hand discomfort after only a short period of playing immediately tells me I need to re-check HOW I'm playing (9/10 times I've gotten lazy and started slouching and playing incorrectly).
Endurance will come with practice and warm ups like everyone has mentioned, but don't underestimate the importance of the correct "playing position".
